A) 0
B) 1
C) 2
D) infinite
Correct Answer: D
Solution :
Key Idea: If system of equations is a homogeneous equation and its determinant is zero, then the solution of the given equation is infinitely. Given system of equations are\[~x+y-\text{ }z=0,\] and \[3x\text{ }-\text{ }y\text{ }-\text{ }z=0\] Now, \[\Delta =\left| \begin{matrix} 1 & 1 & -1 \\ 3 & -1 & -1 \\ 1 & -3 & 1 \\ \end{matrix} \right|\] \[=1(-1-3)-1(3+1)-1(-9+1)\] \[=-4-4+8=0\] \[\therefore \] It has infinite solution. Note: It\[\Delta \ne 0,\] then the only solution is \[x=y=z=0.\]
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